Two Linear Decentralized Procedures

Abstract

We start with a study of the piece-wise linear function determining how the maximum of the objective varies with the right-hand members of the constraints. Generation of this function needs dual prices. The latter being used only in their validity sets, the two proposed procedures avoid the main difficulty to which the Kornar-Liptak procedure is subject. Basically, the subsystems propose at each step new production processes out of various projects which are often not well specified. The central agency sends prices or resource allocations. TwO LINEAR DECENTRALIZED procedures are proposed for finding the optimum of linear programming models. The first procedure operates through prices sent by the central agency to the subsystems; the second one uses resource allocations. The paper starts from the basic idea developed by Kornal and Liptak. That leads to a study of the piece-wise linear function which determines how the maximum of the objective function varies when the right-hand members of the constraints vary. For generating this function, the subsystems report a vector of dual prices to the central agency. These dual prices constitute efficiency indicators for the subsystems, namely marginal gains, only if the conditions of economic activity, set out by the central agency, do not vary too much. With dual prices, the subsystems will provide a validity set defining the set of allocations out of which the dual prices should be revised. This twofold information is equivalent to providing the vector of the coefficients of a production process which the considered subsystem intends to use henceforward (this vector will be the column vector entering the basic matrix). The central agency selects the best among the production processes proposed during the previous iterations. The subsystems are specialized in this way by the central agency. In the first algorithm, this specialization process is carried out by solving a program whereby a certain number of previous propositions are compared against each other. In the second algorithm, specialization is progressive whereas the central agency moves in the allocation space along the reduced gradient direction, that is according to the locally most efficient direction at each iteration. The computations carried out by the subsystems are simple and do not require any optimization. They are straightforward in the first algorithm which merely compares gains yielded by the various available production processes. These two procedures were first described in [5].